farmer wants to enclose his pasture which is bordered by a
river. If he uses 750 feet of fencing on three sides of the field, what is the largest area he can
enclose?
A square would give the largest area, 250 feet by 250 feet.
To find the largest area that the farmer can enclose with 750 feet of fencing, we can consider the geometry of the problem.
Let's assume that the pasture is a rectangle, and the river forms one of its sides. Therefore, the farmer needs to fence the other three sides of the rectangle.
Let's label the sides of the rectangle as follows:
- The side adjacent to the river: x feet
- The two sides perpendicular to the river: y feet each
Since the farmer uses 750 feet of fencing on three sides of the field, we can write the equation:
2y + x = 750
To find the largest area, we need to find the values of x and y that maximize the area of the rectangle. The area of a rectangle is given by the formula:
Area = length * width
In this case, the length of the rectangle is y, and the width is x. Therefore, the area of the rectangle can be expressed as:
Area = xy
Now, we need to eliminate one variable in the equation 2y + x = 750, and substitute it in the area equation. Solving for x in terms of y, we get:
x = 750 - 2y
Substituting this expression for x into the area equation, we have:
Area = y(750 - 2y)
To find the largest possible area, we need to find the value of y that maximizes the area. We can achieve this by taking the derivative of the area equation with respect to y, setting it equal to zero, and solving for y.
Let's differentiate the area equation with respect to y:
d(Area)/dy = 750 - 4y
Setting this derivative equal to zero and solving for y:
750 - 4y = 0
4y = 750
y = 750/4
y = 187.5
Now that we have the value of y, we can substitute it back into the equation 2y + x = 750 to find x:
2(187.5) + x = 750
375 + x = 750
x = 375
Therefore, the dimensions of the rectangle that will enclose the largest area with 750 feet of fencing are:
Length (y) = 187.5 feet
Width (x) = 375 feet
To find the largest area, we substitute these values into the area equation:
Area = (187.5)(375)
Area = 70312.5 square feet
Hence, the largest area the farmer can enclose with 750 feet of fencing is 70312.5 square feet.